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Ficha catalográfica elaborada pelo DePT da
Biblioteca Comunitária da UFSCar
C319cc
Carrocine, Roberta Camelucci.
Caracterizações da compactificação de Poincaré de
campos polinomiais do plano / Roberta Camelucci
Carrocine. -- São Carlos : UFSCar, 2007.
65 f.
Dissertação (Mestrado) -- Universidade Federal de São
Carlos, 2007.
1. Campos vetoriais. 2. Análise matemática. 3. Campos
polinomiais. I. Título.
CDD: 515.73 (20
a
)
Banca Examinadora:
. tfQ~~,
P.t::Dr.José Ruidival Soares
DM-UFSCar
Prof. Dr. Hildeb'tSn'dé Munhoz Rodrigues
ICMC- USP
R
n
Y(x) = (1, 0, ..., 0) R
n
Y
f(x, y) = e
x
cos y
X
f
(x, y) = (−e
x
cos y, −e
x
y)
f R
2
Y
R
n
R
n
Π = {y ∈ R
n+1
| y
n+1
= 1} S
n
H
+
= {y ∈ S
n
| y
n+1
> 0} S
n
Π
φ
+
H
+
R
n
H
−
R
n
R
n
S
n
X R
n
H
+
φ
+
(Dφ
+
)X C
∞
H
+
(Dφ
−
)X φ
−
H
−
C
∞
S
n
X
X
X = (P
1
, ..., P
n
)
m R
n
S
n
X(y) =
1 − y
2
1
−y
1
y
2
... −y
1
y
n
−y
2
y
1
1 − y
2
2
... −y
2
y
n
−y
1
y
n
−y
2
y
n
... 1 − y
2
n
−y
1
y
n+1
−y
2
y
n+1
... −y
n
y
n+1
A
P
1
(y)
P
2
(y)
P
n
(y)
,
P
i
(y
1
, ..., y
n+1
) = y
m
n+1
P
i
(y
1
/y
n+1
, ..., y
n
/y
n+1
)
S
n
R
n+1
R
n
(y
1
, ..., y
n
, y
n+1
) ∈ S
n
i = 1, ..., n+1 1−y
2
i
=
n+1
j=1
j=i
y
2
j
S
n
A =
n
i=2
y
2
i
−y
1
y
2
... −y
1
y
n
−y
2
y
1
n
i=1
i=2
y
2
i
... −y
2
y
n
−y
1
y
n
−y
2
y
n
...
n−1
i=1
y
2
i
−y
1
y
n+1
−y
2
y
n+1
... −y
n
y
n+1
B
X(y) = B.
P (y)
P (y) = (
P
1
(y),
P
2
(y), · · · ,
P
n
(y))
n = 3
π
i
π
i
: R
n
−→ R π
i
(x) = x
i
x =
(x
1
, ..., x
n
)
e
i
i f : R
n
−→ R
∂
e
i
f(x) f i
x ∈ R
n
∂f
∂x
i
(x)
ϕ X ϕ
X
S (ϕ) = ϕ
−1
(R \ {0 })
δ
ij
=
1,
i = j
0, i = j
α = (α
1
, ..., α
n
) n α
n
|α| =
n
i=1
α
i
, α! = α
1
!α
2
!...α
n
!
∂
α
=
∂
|α|
∂x
α
1
1
...∂x
α
n
n
U ⊂ R
n
f : U −→ R f C
k
U f C
k
k
∂
α
f U |α| ≤ k f
C
0
f f : U −→ R
m
U ⊂ R
n
f
C
k
f
i
= π
i
◦f i = 1, ..., m
C
k
f ∈ C
∞
f ∈ C
k
k ≥ 0
M M
n M
R
n
ϕ U ⊂ M R
n
ϕ
ϕ
i
= π
i
◦ ϕ
(U, ϕ) (U, ϕ
1
, ..., ϕ
n
)
(U, ϕ) ϕ(U)
R
n
x ∈ U ϕ(x) = 0
x
M n
F C
k
1 ≤ k ≤ ∞
M
{(U
α
, ϕ
α
)| α ∈ A} A
M =
α∈A
U
α
ϕ
α
◦ ϕ
−1
β
∈ C
k
∀α, β ∈ A
F (U, ϕ)
ϕ
α
◦ ϕ
−1
ϕ ◦ ϕ
−1
α
C
k
α ∈ A (U, ϕ) ∈ F
ϕ
α
: U
α
−→ R
n
ϕ
β
: U
β
−→ R
n
U = U
α
∩ U
β
ϕ
α
◦ ϕ
−1
β
: ϕ
β
(U) −→ ϕ
α
(U)
ϕ
β
(U), ϕ
α
(U) ⊂ R
n
ϕ
α
◦ ϕ
−1
β
∈ C
k
F
0
= {(U
α
, ϕ
α
)| α ∈ A}
F
F
0
n C
k
(M, F)
M n
F C
k
(M, F) M
M
M F
S
n
= {x ∈ R
n+1
|
n+1
i=1
x
2
i
= 1} e
n+1
= (0, ..., 0, 1)
−e
n+1
= (0, .., 0, −1) R
n+1
S
n
F (S
n
\ {e
n+1
}, p
e
n+1
)
(S
n
\ {−e
n+1
}, p
−e
n+1
) p
e
n+1
p
−e
n+1
e
n+1
−e
n+1
U (M, F
M
)
F
U
= {(U
α
∩ U, ϕ
α
(U
α
∩U)
) | (U
α
, ϕ
α
) ∈ F
M
}
(M
1
, F
1
) (M
2
, F
2
) n
1
n
2
(M
1
× M
2
, F) n
1
+ n
2
F
{(U
α
× V
β
, ϕ
α
× ϕ
β
) | (U
α
, ϕ
α
) ∈ F
1
(U
β
, ϕ
β
) ∈ F
2
},
ϕ
α
× ϕ
β
: U
α
× V
β
−→ R
n
1
× R
n
2
C
∞
U ⊂ M M
f : U −→ R C
∞
f ∈ C
∞
(U) f ◦ ϕ
−1
∈ C
∞
ϕ M
ψ : M −→ N M N
C
∞
ψ ∈ C
∞
(M, N) ψ ∈ C
∞
g ∈ C
∞
N g ◦ ψ C
∞
ψ
−1
( g) ψ ∈ C
∞
ϕ ◦ ψ ◦ τ
−1
C
∞
τ M ϕ N
ψ : M −→ N
C
∞
m ∈ M U m
ψ|
U
∈ C
∞
R
n
{U
α
} M
W ⊂ M W ⊂
U
α
U
α
α {U
α
}
{U
α
} W
W {V
β
} {U
α
} β
α V
β
⊂ U
α
{A
α
} M m ∈ M
W
m
m W
m
∩A
α
= ∅ α
M {ϕ
i
; i ∈ I}
C
∞
M
{S (ϕ
i
) ; i ∈ I }
i∈I
ϕ
i
(x) = 1
ϕ
i
(x) ≥ 0 ∀ x ∈ M i ∈ I
{ϕ
i
; i ∈ I} {U
α
}
i α S (ϕ
i
) ⊂ U
α
X
X
ϕ ∈ C
∞
R
n
ϕ ≡ 1
1 C(1) ϕ ≡ 0 C(2)
c
M
{U
α
; α ∈ A} M
{ϕ
i
; i ∈ N} {U
α
} S (ϕ
i
) i
{ϕ
α
}
{U
α
} ϕ
α
G M A M A ⊂ G
ϕ : M −→ R C
∞
0 ≤ ϕ(x) ≤ 1 ∀ x ∈ M
ϕ ≡ 1 A
S (ϕ
i
) ⊂ G
v = (v
1
, ..., v
n
) ∈ R
n
f : R
n
−→ R
p ∈ R
n
v f v(f) f
v p v : C
∞
(R
n
, R) −→ R
v(f) = v
1
∂f
∂x
1
(p) + ... + v
n
∂f
∂x
n
(p)
v(f + λg) = v(f ) + λv(g)
v(fg) = f(p)v(g) + g(p)v(f)
f g p λ
v v
M m ∈ M f g
m f g
m
C
∞
m
˜
F
m
˜
F
m
R
F
m
⊂
˜
F
m
m F
m
˜
F
m
F
k
m
k
k F
m
˜
F
m
⊃ F
m
⊃ F
2
m
⊃ ...
v M m
˜
F
m
f, g ∈
˜
F
m
λ ∈ R
v(f + λg) = v(f) + λ v(g)
v(fg) = f(m) v(g) + g(m) v(f )
M
m
M m
M m v, w ∈ M
m
λ ∈ R
(v + w)(f)
.
= v(f) + w(f ) (λ v)(f)
.
= λ v(f),
v + w λ v m M
m
M
m
M
c c
m v ∈ M
m
v(c) = 0
M
m
(F
m
/F
2
m
)
∗
(F
m
/F
2
m
)
∗
F
m
/F
2
m
p ∈ R
n
g C
k
k ≥ 2
U p q ∈ U
g(q) = g(p) +
n
i=1
∂g
∂x
i
(p)(π
i
(q) − π
i
(p))+
i,j
(π
i
(q) − π
i
(p)) (π
j
(q) − π
j
(p))
1
0
(1 − t)
∂
2
g
∂x
i
∂x
j
(p + t(q − p))dt
g ∈ C
∞
F
2
p
q C
∞
g
g ∈ C
k
k ≥ 2 g ∈ C
2
M m ∈ M F
m
/F
2
m
M F
m
/F
2
m
(U, ϕ) U m
M = n ϕ = (ϕ
1
, ..., ϕ
n
)
˜
F
m
C
∞
F
m
⊂
˜
F
m
f ∈ F
m
f ∈ C
∞
f ◦ ϕ
−1
∈ C
∞
ϕ(m) R
n
ϕ
x m ϕ(x) ϕ(m)
g = f ◦ ϕ
−1
p = ϕ(m) x
m
f(x) = (f ◦ ϕ
−1
)(ϕ(x))
= (f ◦ ϕ
−1
)(ϕ(m)) +
n
i=1
∂(f ◦ ϕ
−1
)
∂x
i
(ϕ(m))
(π
i
(ϕ(x)) − π
i
(ϕ(m)))
+ h
i,j
(π
i
(ϕ(x)) − π
i
(ϕ(m))) (π
j
(ϕ(x)) − π
j
(ϕ(m))),
h ∈ C
∞
(f ◦ ϕ
−1
)(ϕ(m)) = 0 f ∈ F
m
π
i
(ϕ(x)) − π
i
(ϕ(m)) =
ϕ
i
(x) − ϕ
i
(m)
f =
n
i=1
∂(f ◦ ϕ
−1
)
∂x
i
(ϕ(m))
(ϕ
i
− ϕ
i
(m)) + h
i,j
(ϕ
i
− ϕ
i
(m)) (ϕ
j
− ϕ
j
(m))
F
2
m
f =
n
i=1
∂(f ◦ ϕ
−1
)
∂x
i
(ϕ(m))
(ϕ
i
− ϕ
i
(m))
F
2
m
H =
{ϕ
i
− ϕ
i
(m)}; i = 1, ..., n
F
m
/F
2
m
(F
m
/F
2
m
) ≤ n
H
n
i=1
a
i
(ϕ
i
− ϕ
i
(m)) ∈ F
2
m
n
i=1
a
i
(ϕ
i
− ϕ
i
(m))
◦ ϕ
−1
=
n
i=1
a
i
π
i
− π
i
(ϕ(m))
,
n
i=1
a
i
π
i
− π
i
(ϕ(m))
∈ F
2
ϕ(m)
h ∈ F
2
ϕ(m)
h =
n
j=1
f
j
g
j
f
j
, g
j
∈ F
ϕ(m)
f
j
(ϕ(m)) = g
j
(ϕ(m)) = 0
∂
∂x
i
∂h
∂x
i
ϕ(m)
=
n
j=1
∂(f
j
g
j
)
∂x
i
ϕ(m)
=
n
j=1
f
j
(ϕ(m)).
∂g
j
∂x
i
ϕ(m)
+ g
j
(ϕ(m)).
∂f
j
∂x
i
ϕ(m)
= 0
j 1 ≤ j ≤ n
a
j
=
∂
∂x
j
ϕ(m)
n
i=1
a
i
π
i
− π
i
(ϕ(m))
= 0
H F
m
/F
2
m
(F
m
/F
2
m
) = n = M
M
m
= M
M
f m ∈ M
v ∈ M
m
v(f) = v(f ), f ∈
˜
F
m
f m.
v(f) = v(g) f g m
M n m ∈ M (U, ϕ)
m ∈ U ϕ = (ϕ
1
, ..., ϕ
n
) i 1 ≤ i ≤ n
∂
∂ϕ
i
m
∈ M
m
∂
∂ϕ
i
m
(f) =
∂(f ◦ ϕ
−1
)
∂x
i
ϕ(m)
,
f C
∞
m
M
m
f m
ϕ
i
∂
∂ϕ
i
ϕ ϕ
i
∂
∂ϕ
i
m
(f) =
∂f
∂ϕ
i
m
∂
∂ϕ
i
m
; i = 1, ..., n
M
m
{ϕ
i
− ϕ
i
(m)}; i = 1, ..., n
F
m
/F
2
m
∂
∂ϕ
i
m
(ϕ
j
− ϕ
j
(m)) =
∂
(ϕ
j
− ϕ
j
(m)) ◦ ϕ
−1
∂x
i
ϕ(m)
=
∂
(π
j
◦ ϕ − π
j
◦ ϕ(m)) ◦ ϕ
−1
∂x
i
ϕ(m)
=
∂(π
j
− π
j
(ϕ(m)))
∂x
i
ϕ(m)
= δ
ij
v ∈ M
m
v =
n
i=1
a
i
∂
∂ϕ
i
m
v(ϕ
j
) = v(ϕ
j
− ϕ
j
(m)) =
n
i=1
a
i
∂
∂ϕ
i
m
(ϕ
j
− ϕ
j
(m)) = a
j
v =
n
i=1
v(ϕ
i
)
∂
∂ϕ
i
m
M N ψ : M −→ N
C
∞
m ∈ M ψ m
dψ : M
m
−→ N
ψ(m)
,
v ∈ M
m
dψ(v) ∈ N
ψ(m)
dψ(v)(g) = v(g ◦ ψ),
g ∈ C
∞
ψ(m) (g ◦ ψ) ∈ C
∞
g ψ
ψ m dψ
M
m
dψ
m
ψ m dψ
m
(dψ)
f : M −→ R C
∞
v ∈ M
m
f(m) = t
0
df(v) = v(f)
d
dt
t
0
f m
M
∗
m
R
δψ : N
∗
ψ(m)
−→ M
∗
m
ω ∈ N
∗
ψ(m)
δψ(ω) ∈ M
∗
m
δψ(ω)(v) = ω(dψ(v)),
v ∈ M
m
M
∗
m
M m
ψ : M −→ N (U, ϕ
1
, ..., ϕ
n
) (V, σ
1
, ..., σ
k
) m
ψ(m)
∂
∂ϕ
j
m
∈ M
m
∂
∂σ
i
ψ(m)
; i = 1, ..., k
N
ψ(m)
dψ
∂
∂ϕ
j
m
∈ N
ψ(m)
dψ
∂
∂ϕ
j
m
=
k
i=1
dψ
∂
∂ϕ
j
m
(σ
i
)
∂
∂σ
i
ψ(m)
=
k
i=1
∂(σ
i
◦ ψ)
∂ϕ
j
m
∂
∂σ
i
ψ(m)
∂(σ
i
◦ψ)
∂ϕ
j
i=1,...,n
j=1,...,k
ψ
(U, ϕ
1
, ..., ϕ
n
) M m ∈ U
{dϕ
i
|
m
; i = 1, ..., n} M
∗
m
∂
∂ϕ
i
m
; i = 1, ..., n
M
m
f : M −→ R C
∞
v ∈ M
m
df(v) = v(f)
d
dt
f(m)
ϕ
i
: U ⊂ M −→ R ϕ
i
∈ C
∞
∂
∂ϕ
i
m
∈ M
m
dϕ
i
∂
∂ϕ
j
m
=
∂
∂ϕ
j
m
(ϕ
i
)
d
dt
ϕ
i
(m)
∂
∂ϕ
j
m
(ϕ
i
) =
∂(ϕ
i
◦ ϕ
−1
)
∂x
j
ϕ(m)
=
∂((π
i
◦ ϕ) ◦ ϕ
−1
)
∂x
j
ϕ(m)
=
∂π
i
∂x
j
ϕ(m)
= δ
ij
,
dϕ
i
∂
∂ϕ
j
m
= δ
ij
f : M −→ R C
∞
df
m
∈ M
∗
m
df
m
∂
∂ϕ
j
m
=
n
i=1
a
i
dϕ
i
∂
∂ϕ
j
m
= a
j
⇒ a
j
= df
m
∂
∂ϕ
j
m
=
∂f
∂ϕ
j
m
df
m
=
n
i=1
∂f
∂ϕ
j
|
m
dϕ
i
m
ψ : M −→ N ϕ : N −→ X C
∞
d(ϕ ◦ ψ)
m
= dϕ
ψ(m)
◦ dψ
m
d(ϕ ◦ ψ)
m
: M
m
−→ X
ϕ(ψ(m))
v ∈ M
m
g ∈ C
∞
(ϕ ◦ ψ)(m)
d(ϕ ◦ ψ)
m
(v)(g) = v(g ◦ (ϕ ◦ ψ)) = v((g ◦ ϕ) ◦ ψ)
= dψ
m
(v)(g ◦ ϕ) = dϕ
ψ(m)
(dψ
m
(v))(g)
d(ϕ ◦ ψ)
m
= dϕ
ψ(m)
◦ dψ
m
ψ : M −→ N f : N −→ R C
∞
δψ(df
ψ(m)
) = d(f ◦ ψ)
m
ψ : M −→ N ∈ C
∞
M N
dψ
m
≡ 0 ∀ m ∈ M ψ
ψ : M −→ N C
∞
ψ dψ
m
m ∈ M
(M, ψ) N ψ
ψ ψ ψ : M −→ ψ(M)
ψ(M) N
ψ ψ ψ
−1
∈ C
∞
(U, ϕ) M
ϕ : U −→ ϕ(U) ψ : M −→ N
dψ
m
m
dψ
m
ψ
m
{ϕ
1
, ..., ϕ
j
} C
∞
m ∈ M m dϕ
1
m
, ..., dϕ
j
m
M
∗
m
M n {ϕ
1
, ..., ϕ
j
} m
j ≤ n
U ⊂ R
n
f : U −→ R
n
C
∞
∂(π
i
◦f)
x
j
i,j=1,...,n
x ∈ U V ⊂ U
x ∈ V f|
V
: V −→ f (V )
ψ : M −→ N C
∞
m ∈ M dψ : M
m
−→ N
ψ(m)
U m ψ : U −→ ψ( U)
ψ(U) ⊂ N
dψ
m
M = M
m
= N
ψ(m)
= N
M = n (V, ϕ) m (W, σ) ψ(m)
ψ(V ) ⊂ W ϕ(m) = p σ( ψ(m)) = q
(σ ◦ ψ ◦ ϕ
−1
)|
ϕ(V )
: ϕ(V ) −→ σ(W )
d(σ ◦ ψ ◦ ϕ
−1
)
p
: ϕ(V )
p
−→ σ(W )
q
p
dψ
m
ϕ σ
ϕ : V −→ ϕ(V ) σ : W −→ σ(W )
dσ
ψ(m)
dϕ
−1
p
d(σ ◦ ψ ◦ ϕ
−1
)
p
= dσ
ψ(m)
◦ d(ψ ◦ ϕ
−1
)
p
= dσ
ψ(m)
◦ dψ
m
◦ (dϕ
−1
)
p
,
d(σ ◦ ψ ◦ ϕ
−1
)
p
h = σ ◦ ψ ◦ ϕ
−1
ϕ(V ) R
n
σ(W ) ⊂ R
n
C
∞
dh
p
˜
U ⊂ ϕ(V ) p h :
˜
U −→ h(
˜
U)
ϕ σ ϕ(V ) σ(W ) ϕ
−1
(
˜
U) ⊂ V
˜
U ⊂ ϕ(V )
(σ
−1
◦ h|
˜
U
◦ ϕ) : ϕ
−1
(
˜
U) −→ σ
−1
(h(
˜
U))
U = ϕ
−1
(
˜
U)
σ
−1
◦ h|
˜
U
◦ ϕ = (σ
−1
◦ h ◦ ϕ)
ϕ
−1
(
˜
U)
= (σ
−1
◦ σ ◦ ψ ◦ ϕ
−1
◦ ϕ)
U
= ψ|
U
ψ(U) = σ
−1
(h(
˜
U)) ψ : U −→ ψ(U)
M = n {ϕ
1
, ...ϕ
n
}
m
0
∈ M {ϕ
1
, ...ϕ
n
}
m
0
M = n σ
1
, ..., σ
k
k < n
m
m
ψ : M −→ N ψ ∈ C
∞
dψ : M
m
−→ N
ψ(m)
(σ
1
, ..., σ
k
) ψ(m)
(σ
1
◦ ψ), ..., (σ
k
◦ ψ)
m
σ
1
, ..., σ
k
C
∞
m ∈ M
M
∗
m
{σ
1
, ..., σ
k
}
m
ψ : M −→ N ψ ∈ C
∞
dψ : M
m
−→ N
ψ(m)
(σ
1
, ..., σ
k
) ψ(m)
{(σ
i
◦ ψ); i = 1, ..., k}
m ψ m
ψ : N −→ M C
∞
(P, ϕ) M
ϕ : P −→ M ∈ C
∞
ψ
(P, ϕ) ψ(N) ⊂ ϕ(P )
ψ
0
: N −→ P ϕ ◦ ψ
0
= ψ
(P, ϕ) M ϕ : P −→ ϕ(P )
ψ
0
= ϕ
−1
ϕ(P )
◦ ψ ψ(N) ⊂ ϕ(P )
ψ
0
C
∞
ψ : N −→ M C
∞
(P, ϕ) M
ψ (P, ϕ) ψ
0
: N −→ P ϕ ◦ ψ
0
= ψ
ψ
0
ψ
0
∈ C
∞
ϕ ψ
0
M = n P = k (P, ϕ)
ϕ : P −→ M dϕ
p
p ∈ P
k = P = P
p
≤ M
ϕ(p)
= M = n
ψ
0
∈ C
∞
τ ◦ ψ
0
∈ C
∞
τ P τ ◦ ψ
0
∈ C
∞
τ P
τ ◦ ψ
0
◦ σ
−1
∈ C
∞
σ N
ψ
0
∈ C
∞
P (U, τ)
τ ◦ ψ
0
ψ
−1
0
(U) C
∞
p ∈ P (V, γ) γ = (γ
1
, ..., γ
n
)
ϕ(p) dϕ
p
{γ
i
◦ϕ ; i = 1, ..., n} {(γ
i
1
◦ϕ), . . . , (γ
i
k
◦ϕ)}
U p
π : R
n
−→ R
k
k γ
γ = (γ
i
1
, . . . , γ
i
k
, γ
i
k+1
, . . . , γ
i
n
)
τ = π◦γ ◦ ϕ = (γ
i
1
◦ϕ, . . . , γ
i
k
◦ϕ)
U ϕ ◦ ψ
0
= ψ
(τ ◦ ψ
0
)
ψ
−1
0
(U)
= (π ◦ γ ◦ ϕ ◦ ψ
0
)
ψ
−1
0
(U)
= (π ◦ γ ◦ ψ)
ψ
−1
0
(U)
,
(τ ◦ ψ
0
)
ψ
−1
0
(U)
∈ C
∞
π γ ψ ∈ C
∞
p ∈ P (U, τ) p
(τ ◦ψ
0
)
ψ
−1
0
(U)
∈ C
∞
P
(U, τ) τ ◦ ψ
0
∈ C
∞
ϕ ϕ : P −→ ϕ(P )
U ⊂ P ψ ∈ C
∞
ψ
−1
(ϕ(U))
N
ψ
−1
(ϕ(U)) = (ϕ ◦ ψ
0
)
−1
(ϕ(U)) = ψ
−1
0
(ϕ
−1
(ϕ(U)) ) = ψ
−1
0
(U),
ψ
−1
0
(U) N ψ
0
(N
1
, ϕ
1
) (N
2
, ϕ
2
) M
α : N
1
−→ N
2
ϕ
1
= ϕ
2
◦ α
M
ξ (A, i) A ⊂ M
i : A −→ M C
∞
(N, ϕ)
ξ A = ϕ(N) A
ϕ : N −→ A (A, i)
M (N, ϕ) ϕ = i ◦ ϕ ϕ
i A
A ϕ N = n
ϕ
−1
: A −→ N a ∈ A (U, σ)
ϕ
−1
(a) ϕ(U) A a σ U σ(U)
ϕ
−1
σ ◦ ϕ
−1
: ϕ(U) −→ σ(U)
(ϕ(U), σ ◦ ϕ
−1
) a
F = {(U
α
, σ
α
)| α ∈ J} C
∞
N
N =
α∈J
U
α
σ
α
◦ σ
−1
β
∈ C
∞
∀ α, β ∈ J F
F
′
= {(ϕ(U
α
), σ
α
◦ ϕ
−1
)| α ∈ J}
A = ϕ(N) = ϕ
α∈J
U
α
=
α∈J
ϕ(U
α
)
τ
α
= σ
α
◦ ϕ
−1
∀ α, β ∈ J
τ
α
◦ τ
−1
β
= σ
α
◦ ϕ
−1
◦ (σ
β
◦ ϕ
−1
)
−1
= σ
α
◦ σ
−1
β
τ
α
◦ τ
−1
β
∈ C
∞
(V, τ) A τ
α
◦ τ
−1
τ ◦ τ
−1
α
C
∞
∀ α ∈ J
(V, τ) ∈ F
′
(ϕ
−1
(V ), τ ◦ ϕ) N
(τ ◦ϕ)◦ σ
−1
α
= τ ◦ τ
−1
α
σ
α
◦(τ ◦ ϕ)
−1
= τ
α
◦ τ
−1
C
∞
∀ α ∈ J
F (ϕ
−1
(V ), τ ◦ ϕ) ∈ F (ϕ(ϕ
−1
(V )), (τ ◦ ϕ) ◦ ϕ
−1
) =
(V, τ) ∈ F
′
F
′
A
M A ⊂ M
A A (A, i)
M
A ⊂ M Ψ A A
Ψ F
1
(A, i)
M
F
2
(A, i)
M F
k
= {(U
α
k
, ϕ
α
k
)| α
k
∈ J
k
} k = 1, 2
F
1
= F
2
A
1
= (A, Ψ, F
1
) A
2
= (A, Ψ, F
2
) (A
1
, i
1
) (A
2
, i
2
)
M A
1
A
2
id : A
1
−→ A
2
ϕ
α
2
◦ id ◦ ϕ
−1
α
1
= ϕ
α
2
◦ ϕ
−1
α
1
ϕ
α
1
◦ id
−1
◦ ϕ
−1
α
2
= ϕ
α
1
◦ ϕ
−1
α
2
C
∞
F
1
F
2
F
1
= F
2
A (A, i)
M
M A ⊂ M
A (A, i) M A
(A, i) M
A
ψ : M −→ N C
∞
dψ
m
m ∈ M ψ
Ψ
1
A
F
1
(A, i)
M A
1
= (A, Ψ
1
, F
1
)
(A, i)
M Ψ
2
F
2
A
2
= (A, Ψ
2
, F
2
)
(A
1
, i
1
) (A
2
, i
2
) M i
1
(A) = i
2
(A) = A i
1
, i
2
∈ C
∞
i
1
i
2
i
1
◦ id = i
2
id id : A
2
−→ A
1
A
1
i
1
id ∈ C
∞
i
1
i
2
di
1
−1
di
2
a ∈ A id = i
1
−1
◦ i
2
d(id) = d i
1
−1
◦ d i
2
a ∈ A id
Ψ
1
Ψ
2
F
1
= F
2
A (A, i)
M
(U, ϕ) ϕ = (ϕ
1
, ..., ϕ
n
) M
k ∈ Z 0 ≤ k < n x ∈ ϕ(U)
S = {m ∈ U | ϕ
i
(m) = π
i
(x), i = k + 1, ..., n}
S M {ϕ
j
|
S
; j = 1, ..., k}
M (U, ϕ)
S = k
ψ : M −→ N m ∈ M
(V, ϕ) ψ(m) U m ψ|
U
ψ(U) (V, ϕ)
(M, ψ) N f ∈ C
∞
(N) f ◦ψ ∈ C
∞
(M)
ψ : M −→ N ψ(M) N
g ◦ ψ ∈ C
∞
(M) f ◦ ψ ∈ C
∞
(N) f ◦ ψ = g
U ⊂ R
n−k
×R
k
f : U −→ R
k
C
∞
(x, y) = (x
1
, ..., x
n−k
, y
1
, ..., y
k
) R
n−k
× R
k
(x
0
, y
0
) ∈ U
f(x
0
, y
0
) = 0
∂f
i
∂y
j
(x
0
,y
0
)
i,j=1,...,k
V x
0
R
n−k
W y
0
R
k
V × W ⊂ U g : V −→ W ∈ C
∞
(x, y) ∈ V × W
f(x, y) = 0 ⇐⇒ g(x) = y
M N n k
ψ : M −→ N ∈ C
∞
x ∈ N P = ψ
−1
(x)
dψ : M
m
−→ N
ψ(m)
m ∈ M P
(P, i) i m
i P = n − k
P
(P, i) M P = n −k
P m ∈ P
U m P ∩ U
n − k
(ϕ
1
, ..., ϕ
k
) x ∈ N m ∈ P = ψ
−1
(x)
dψ
m
{σ
i
= (ϕ
i
◦ ψ); i = 1, ..., k} {σ
1
, ..., σ
k
,
σ
k+1
, ..., σ
n
} U m
p ∈ P ∩ U i = 1, ..., k σ
i
(p) = ϕ
i
(ψ(p)) = ϕ
i
(x) = 0
(ϕ
1
, ..., ϕ
k
) x P ∩ U = {p ∈ U | σ
i
(p) = 0; i = 1, ..., k}
P ∩ U (U, σ
1
, ..., σ
k
, σ
k+1
, ..., σ
n
)
(P, i) M n − k
C
∞
0
ψ : M −→ N ∈ C
∞
(O, ϕ) N
P = ψ
−1
(ϕ(O)) m ∈ ψ
−1
(ϕ(O))
N
ψ(m)
= dψ(M
m
) + dϕ(O
ϕ
−1
(ψ(m))
),
P (P, i) M
P = M − N + O.
(O, ϕ) (P, i)
P (P, i) M
(a, b) M
M σ : (a, b) −→ M ∈ C
∞
t
0
∈ (a, b)
dσ
d
dt
t
0
∈ M
σ(t
0
)
σ t
0
˙σ(t
0
)
v = 0 M
m
(U, ϕ) m v = dϕ
−1
∂
dx
1
0
v
t −→ ϕ
−1
(t, 0, ..., 0) 0
σ : [a, b] −→ M M
σ C
∞
(a − ǫ, b + ǫ) M ǫ > 0
σ : [a, b] −→ M a = α
0
< α
1
< ... <
α
n
= b
σ
[α
i
,α
i+1
]
i = 0, ..., n − 1
σ : [a, b] −→ M M ˙σ(t
0
) = dσ
d
dt
t
0
∈
M
σ(t
0
)
t
0
∈ [a, b]
M
M
T (M) =
m∈M
M
m
T
∗
(M) =
m∈M
M
∗
m
π : T (M) −→ M π
∗
: T
∗
(M) −→ M
π(v) = m, v ∈ M
m
π
∗
(ω) = m, ω ∈ M
∗
m
(U, ϕ) ϕ = (ϕ
1
, ..., ϕ
n
) M
ϕ : π
−1
(U) −→ R
2n
ϕ
∗
: (π
∗
)
−1
(U) −→ R
2n
ϕ(v) =
ϕ
1
(π(v)), . . . , ϕ
n
(π(v)), dϕ
1
(v), . . . , dϕ
n
(v)
ϕ
∗
(ω) =
ϕ
1
(π
∗
(ω)), . . . , ϕ
n
(π
∗
(ω)), ω(
∂
∂ϕ
1
), . . . , ω(
∂
∂ϕ
n
)
F M ϕ ϕ
∗
(U, ϕ) (V, ψ) ∈ F
ψ ◦ ϕ
−1
ψ
∗
◦ (ϕ
∗
)
−1
C
∞
T (M) T
∗
(M)
{ϕ
−1
(W ) | W R
2n
, (U, ϕ) ∈ F} {(ϕ
∗
)
−1
(W ) | W R
2n
,
(U, ϕ) ∈ F} T (M) T
∗
(M)
2n
F
F
∗
{(π
−1
(U), ϕ) | (U, ϕ) ∈ F} {((π
∗
)
−1
(U), ϕ
∗
) | (U, ϕ) ∈ F}
F
F
∗
T (M) T
∗
(M)
X σ : [a, b] −→ M
X : [a, b] −→ T(M) σ π ◦ X = σ X(t) ∈ M
σ(t)
X ∈ C
∞
X U ⊂ M U T(M)
X : U −→ T(M) π ◦ X = id
U
X(m) ∈ M
m
X X ∈ C
∞
(U, T(M)) X(m) = X
m
X U f ∈ C
∞
(U) X(f) : U −→ R
X(f)(m) = X
m
(f)
A = {X ∈ C
∞
(U, T(M)) | X } A
R C
∞
(U)
R × A A (f, X) −→ fX fX : U −→ T(M)
(fX)(m) = f(m)X(m)
X M
X ∈ C
∞
(U, ϕ) ϕ = (ϕ
1
, ..., ϕ
n
) M
{a
i
; i = 1, ..., n} U
X
U
=
n
i=1
a
i
∂
∂ϕ
i
,
a
i
∈ C
∞
(U)
V M f ∈ C
∞
(V ) X(f) ∈ C
∞
(V)
⇒
(U, ϕ) = (U, ϕ
1
, ..., ϕ
n
) M X
C
∞
{a
i
; i = 1, ..., n} U
X
U
=
n
i=1
a
i
∂
∂ϕ
i
X
U
∈ C
∞
(π
−1
(U), ϕ) T(M)
dϕ
i
: π
−1
(U) −→ R ∈ C
∞
dϕ
i
ϕ
dϕ
i
◦ X
U
∈ C
∞
dϕ
i
◦ X
U
= a
i
a
i
∈ C
∞
⇒
V ⊂ M f ∈ C
∞
(V ) X(f) ∈ C
∞
(V)
(U, ϕ
1
, ..., ϕ
n
) M U ⊂ V
X(f)
U
∈ C
∞
(U)
(U, ϕ
1
, ..., ϕ
n
) m ∈ U
X
U
(m) = (X
U
)
m
∈ M
m
U
X(f)(m) = X
m
(f) =
n
i=1
X
m
(ϕ
i
)
∂f
∂ϕ
i
m
f ∈ C
∞
(V ) f ∈ C
∞
(U)
∂f
∂ϕ
i
∈ C
∞
a
i
= (X
U
)(ϕ
i
)
a
i
∈ C
∞
X(f)
U
∈ C
∞
C
∞
(U, ϕ
1
, ..., ϕ
n
) M
U ⊂ V X(f) ∈ C
∞
(V)
⇒
X ∈ C
∞
(U, ϕ) = (U, ϕ
1
, ..., ϕ
n
) M ϕ ◦ X|
U
∈ C
∞
ϕ
(π
−1
(U), ϕ) ϕ ◦ X|
U
∈ C
∞
ϕ ◦ X|
U
◦ ϕ
−1
∈ C
∞
X|
U
∈ C
∞
(U, ϕ) X|
U
∈ C
∞
X ∈ C
∞
ϕ =
(ϕ
1
◦ π), . . . , (ϕ
n
◦ π), dϕ
1
, . . . , dϕ
n
ϕ ◦ X|
U
=
(ϕ
1
◦ π ◦ X|
U
), ..., (ϕ
n
◦ π ◦ X|
U
), (dϕ
1
◦ X|
U
), ..., (dϕ
n
◦ X|
U
)
π ◦ X|
U
= id
U
X ϕ
i
◦ π ◦ X|
U
= ϕ
i
∈ C
∞
m ∈ U X(m) = X
m
∈ M
m
X
m
=
n
i=1
X
m
(ϕ
i
)
∂
∂ϕ
i
m
=
n
i=1
X(ϕ
i
)(m)
∂
∂ϕ
i
m
(dϕ
j
◦ X|
U
)(m) = dϕ
j
n
i=1
X(ϕ
i
)(m)
∂
∂ϕ
i
m
=
n
i=1
X(ϕ
i
)(m) dϕ
j
∂
∂ϕ
i
m
= X(ϕ
j
)(m)
ϕ
j
∈ C
∞
(U) X(ϕ
j
) ∈ C
∞
(U) dϕ
j
◦ X|
U
∈ C
∞
(U)
ϕ
j
◦ π ◦ X|
U
dϕ
j
◦ X|
U
∈ C
∞
˜ϕ ◦ X|
U
∈ C
∞
⇒
X Y M
[X, Y] M m ∈ M [X, Y]
m
∈ M
m
[X, Y]
m
(f) = X
m
(Yf) − Y
m
(Xf),
f ∈ C
∞
m [X, Y]
X Y
[X, Y]
m
(f + λg) = [X, Y]
m
(f) + λ [X, Y]
m
(g) [X, Y]
m
(fg) =
f(m) [X, Y]
m
(g) + g(m) [X, Y]
m
(f) [X, Y]
m
∈ M
m
[X, Y]
X Y Z M f, g ∈ C
∞
(M)
[X, Y] M
[fX, gY] = fg[X, Y] + f(Xg)Y − g(Yf)X
[X, Y] = −[Y, X]
[[X, Y], Z] + [[Y, Z], X] + [[Z, X], Y] = 0
[X + Y, Z] = [X, Z] + [Y, Z] [X, Y + Z] = [X, Y] + [X, Z]
A
X M σ M
X
˙σ(t) = X(σ(t))
t σ
X m ∈ M
X m γ : (a, b) −→ M
X
˙γ(t) = dγ
d
dt
t
= X(γ(t)), t ∈ (a , b)
0 ∈ (a, b) γ(0) = m (U, ϕ) ϕ = (ϕ
1
, ..., ϕ
n
)
m
X
U
=
n
i=1
f
i
∂
∂ϕ
i
,
f
i
= X(ϕ
i
) X ∈ C
∞
f
i
∈ C
∞
t γ(t) ∈ U
dγ
d
dt
t
=
n
i=1
d(ϕ
i
◦ γ)
dt
t
∂
∂ϕ
i
γ(t)
γ X
n
i=1
d(ϕ
i
◦ γ)
dt
t
∂
∂ϕ
i
γ(t)
=
n
i=1
f
i
(γ(t))
∂
∂ϕ
i
γ(t)
γ
i
= ϕ
i
◦ γ γ X γ
−1
(U)
dγ
i
dt
t
= f
i
◦ ϕ
−1
(γ
1
(t), ..., γ
n
(t)) (i = 1, ..., n
t ∈ γ
−1
(U))
i = 1, ..., n
X M
m ∈ M a(m), b(m) ∈ R ∪ {+∞, −∞}
γ
m
: (a(m), b(m)) −→ M
0 ∈ (a(m), b(m)) γ
m
(0) = m
γ
m
X
µ : (c, d) −→ M X
(c, d) ⊂ (a(m), b(m)) µ = γ
m
(c,d)
t ∈ R D
t
= {m ∈ M | t ∈ (a(m), b(m))}
Y
t
: D
t
−→ M Y
t
(m) = γ
m
(t)
m ∈ M V m ǫ > 0
σ : (−ǫ, ǫ) × V −→ M σ(t, x) = Y
t
(x) C
∞
D
t
t ∈ R
t>0
D
t
= M
Y
t
: D
t
−→: D
−t
Y
−t
s, t ∈ R (Y
s
◦ Y
t
) ⊂ D
s+t
t s
(Y
s
◦ Y
t
) = D
s+t
Y
s
◦ Y
t
= Y
s+t
(Y
s
◦ Y
t
)
X M D
t
= M
t ∈ R m ∈ M γ
m
(−∞, +∞)
Y
t
M
X
ψ : M −→ N ∈ C
∞
X : M −→ T(N)
ψ X ∈ C
∞
(M, T(N)) π ◦ X = ψ
X C
∞
N m ∈ M U m V
ψ(m) ψ(U) ⊂ V
X : V −→ T (N)
X ◦ ψ|
U
= X|
U
M n m ∈ M X
M X(m) = 0 0
(U, ϕ) ϕ = (ϕ
1
, ..., ϕ
n
) m U
X =
∂
∂ϕ
1
ϕ : M −→ N ∈ C
∞
X Y M N
X Y ϕ
dϕ ◦ X = Y ◦ ϕ
ϕ : M −→ N ∈ C
∞
X X
1
M Y Y
1
N X ϕ Y X
1
ϕ Y
1
[X, X
1
] [Y, Y
1
] ϕ
M n k ∈ Z
+
1 ≤ k ≤ n k D M
k D(m) M
m
m ∈ M
D m ∈ M U m k
X
1
, ..., X
k
C
∞
U D U
x ∈ U D(x) X
1
(x), ..., X
k
(x)
X M D X(m) = X
m
∈ D(m)
m ∈ M
D
X, Y ∈ D [X, Y] ∈ D
(N, ψ) M D M
dψ(N
x
) = D(ψ(x)), ∀ x ∈ N,
(N, ψ) D
D M M
D D
X Y D
[X, Y](m) ∈ D(m) m ∈ M
m ∈ M (N, ψ) D m
ψ(x
0
) = m
dψ(N
x
) = D(ψ(x)) ∀ x ∈ N ψ
dψ : N
x
−→ D(ψ(x))
X
Y
N
dψ ◦
X = X ◦ ψ dψ ◦
Y = Y ◦ ψ
dψ
X = dψ
−1
◦ X ◦ ψ
Y = dψ
−1
◦ Y ◦ ψ
X X ψ
Y Y
[
X,
Y] [X, Y] ψ
[X, Y](m) = [X, Y](ψ(x
0
)) = dψ([
X,
Y](x
0
)) ∈ D(m)
[X, Y](m) ∈ D(m) m M
[X, Y] D
D(m) k
M n ≥ k m ∈ M D(m)
m (U, ϕ)
m ϕ = (ϕ
1
, . . . , ϕ
n
)
{ϕ
i
= ; i = k + 1, ..., n}
D(m) (N, ψ) D(m)
ψ(N) ⊂ U ψ(N)
E
ϕ : E −→ F F ϕ(E) F ϕ
E ϕ(E)
F E
ϕ
E
E ϕ : E −→ F
F
ϕ E ϕ(E)
F = ϕ(E) ∪ {∞} ∞ /∈ ϕ(E)
F
E
∞
E x ∈ E
E
x ∈ E V E V
E
E
R
n
Π
0
= {y ∈ R
n+1
| y
n+1
= −1}
S
n
= {y ∈ R
n+1
|
n+1
i=1
y
2
i
= 1}
−e
n+1
=
(0, ..., 0, −1)
S
n
\{e
n+1
} R
n
p
Π
0
e
n+1
p Π
0
R
n
S
n
\ {e
n+1
} ϕ : R
n
−→ S
n
ϕ : R
n
−→ S
n
\ {e
n+1
} ϕ
S
n
R
n
R
n
B
1
(0) R
n
B
1
(0) R
n
B
1
(0) R
n
S
n−1
R
n
B
1
(0)
S
n
R
n+1
R
n
R
n
Π = {y ∈ R
n+1
| y
n+1
= 1} S
n
H
+
= {y ∈ S
n
| y
n+1
> 0} S
n
Π
R
n
E
x ∈ E U E x ∈ U f : E −→ [0, 1]
f(x) = 1 f(E \ U) = 0
E I = [0, 1]
F = {f : E −→ I | f } I
F
=
f∈F
I
I
F
ϕ : E −→ I
F
ϕ(x) = {f(x)}
f∈F
ϕ
ϕ : E −→ ϕ(E) E
E β(E) = ϕ(E)
ϕ : E −→ β(E) E
ϕ : E −→ β(E)
E
f : E −→ R f : β(E) −→ R f ◦ϕ = f
S
n
R
n
X = (P
1
, ..., P
n
) R
n
R
n
Π = {y ∈ R
n+1
| y
n+1
= 1} S
n
= {y ∈ R
n+1
|
n+1
i=1
y
2
i
= 1}
e
n+1
= (0, . . . , 0, 1) {y ∈
S
n
| y
n+1
> 0} {y ∈ S
n
| y
n+1
< 0} S
n
H
+
H
−
X X
H
+
H
−
Π
x ∈ R
n
∆(x) = (1 +
n
i=1
x
2
i
)
1/2
φ
+
: R
n
−→ H
+
φ
−
: R
n
−→ H
−
φ
+
(x) =
1
∆(x)
(x
1
, ..., x
n
, 1)
φ
−
(x) = −
1
∆(x)
(x
1
, ..., x
n
, 1)
X
H
+
∪H
−
X
X(y) =
(Dφ
+
)
x
X(x),
y = φ
+
(x)
(Dφ
−
)
x
X(x),
y = φ
−
(x)
φ
+
(x) = (f
1
(x), ..., f
n
(x), f
n+1
(x)) f
i
(x) =
x
i
∆(x)
i = 1, ..., n
f
n+1
(x) =
1
∆(x)
y = φ
+
(x) (y
1
, ..., y
n
, y
n+1
) = (
x
1
∆(x)
, ...,
x
n
∆(x)
,
1
∆(x)
)
(Dφ
+
)
x
= [∂
e
j
f
i
(x)]
i=1,...,n+1
j=1,...,n
∂
e
i
f
i
(x) = ∂
e
i
(
x
i
∆(x)
) =
1
∆(x)
−
x
2
i
∆
3
(x)
= y
n+1
− y
n+1
y
2
i
,
1 ≤ i ≤ n
∂
e
j
f
i
(x) = ∂
e
j
(
x
i
∆(x)
) = −
x
i
x
j
∆
3
(x)
= −y
n+1
y
i
y
j
,
i = j
∂
e
j
f
n+1
(x) = ∂
e
j
(
1
∆(x)
) = −
x
j
∆
3
(x)
= −y
2
n+1
y
j
j = 1, ..., n
(y
1
, ..., y
n
, y
n+1
) = (−
x
1
∆(x)
, ..., −
x
n
∆(x)
, −
1
∆(x)
) y = φ
−
(x) (Dφ
+
)
x
=
(Dφ
−
)
x
X(y) H
+
∪ H
−
X(y) = y
n+1
1 − y
2
1
−y
1
y
2
... −y
1
y
n
−y
2
y
1
1 − y
2
2
... −y
2
y
n
−y
1
y
n
−y
2
y
n
... 1 − y
2
n
−y
1
y
n+1
−y
2
y
n+1
... −y
n
y
n+1
P
1
(y)
P
2
(y)
P
n
(y)
,
P
i
(y
1
, ..., y
n+1
) = P
i
(y
1
/y
n+1
, ..., y
n
/y
n+1
)
φ
+
φ
−
S
n
S
n−1
= {y ∈ S
n
| y
n+1
= 0} R
n
(X) =
max{ (P
1
), ..., (P
n
)} = m
X S
n
\ S
n−1
= H
+
∪ H
−
)
X S
n
X(y) = y
m−1
n+1
X(y) =
1 − y
2
1
−y
1
y
2
... −y
1
y
n
−y
2
y
1
1 − y
2
2
... −y
2
y
n
−y
1
y
n
−y
2
y
n
... 1 − y
2
n
−y
1
y
n+1
−y
2
y
n+1
... −y
n
y
n+1
P
1
(y)
P
2
(y)
P
n
(y)
,
P
i
(y
1
, ..., y
n+1
) = y
m
n+1
P
i
(y
1
/y
n+1
, ..., y
n
/y
n+1
) m
X
X
X S
n
X
X
S
n
\ S
n−1
X y
m−1
n+1
H
+
H
−
m
P
i
(−y) = (−1)
m
P
i
(y) m
X R
n+1
X(−y) = (−1)
m
X(y) = −
X(y)
X
R
n+1
X
X
m
m
X S
n
2(n + 1) (U
i
, φ
+
i
)
(V
i
, φ
−
i
)
i = 1, ..., n + 1
U
i
= {y ∈ S
n
| y
i
> 0}, V
i
= {y ∈ S
n
| y
i
< 0}
φ
+
i
: U
i
−→ R
n
φ
−
i
: V
i
−→ R
n
(n + 1)
U
i
V
i
n (
y
1
y
i
, ...,
y
i−1
y
i
,
y
i+1
y
i
, ...,
y
n+1
y
i
)
(z
1
, ..., z
n
)
n
(U
n+1
, φ
+
n+1
) (V
n+1
, φ
−
n+1
)
z
n
U
n+1
= H
+
V
n+1
= H
−
φ
+
n+1
= (φ
+
)
−1
φ
−
n+1
= (φ
−
)
−1
X
1 ≤ i ≤ n y ∈ U
i
∩ H
+
(Dφ
+
i
)
y
T
y
U
i
T
φ
+
i
(y)
R
n
(Dφ
+
i
)
y
(
X(y)) = (Dφ
+
i
)
y
(y
m−1
n+1
X(y)) = y
m−1
n+1
(Dφ
+
i
)
y
(
X(y)) =
y
m−1
n+1
(Dφ
+
i
)
y
((Dφ
+
)
x
X(x)) = y
m−1
n+1
(D(φ
+
i
◦ φ
+
))
x
X(x),
y = φ
+
(x)
(φ
+
i
◦ φ
+
)(x) = φ
+
i
(
x
1
∆(x)
, . . . ,
x
n
∆(x)
,
1
∆(x)
) = (
x
1
x
i
, . . . ,
x
i−1
x
i
,
x
i+1
x
i
, . . . ,
x
n
x
i
,
1
x
i
)
(D(φ
+
i
◦ φ
+
))
x
(φ
+
i
◦ φ
+
)(x)
(D(φ
+
i
◦ φ
+
))
x
X(x) =
1
x
i
... 0 −
x
1
x
2
i
0 ... 0
0 ...
1
x
i
−
x
i−1
x
2
i
... 0
0 ... 0 −
x
i+1
x
2
i
1
x
i
... 0
0 ... 0 −
x
n
x
2
i
0 ...
1
x
i
0 ... 0 −
1
x
2
i
0 ... 0
P
1
(x)
P
2
(x)
P
n
(x)
=
1
x
2
i
x
i
P
1
−x
1
P
i
, x
i
P
2
−x
2
P
i
, · · · , x
i
P
i−1
−x
i−1
P
i
, x
i
P
i+1
−x
i+1
P
i
, · · · , x
i
P
n
−x
n
P
i
, −P
i
y = φ
+
(x) U
i
(z
1
, ..., z
n
) = (
x
1
x
i
, ...,
x
i−1
x
i
,
x
i+1
x
i
, ...,
x
n
x
i
,
1
x
i
) = φ
+
i
(φ
+
(x)) =
φ
+
i
(y
1
, ..., y
n+1
) = (
y
1
y
i
, ...,
y
i−1
y
i
,
y
i+1
y
i
, ...
y
n+1
y
i
)
D(φ
+
i
◦ φ
+
))
x
X(x) = z
n
P
1
−z
1
P
i
, · · · , P
i−1
−z
i−1
P
i
, P
i+1
−z
i
P
i
, · · · , P
n
−z
n−1
P
i
, −z
n
P
i
,
P
j
= P
j
(
z
1
z
n
, ...,
z
i−1
z
n
,
1
z
n
,
z
i
z
n
, ...,
z
n−1
z
n
)
∆(z) =
1
|y
i
|
y ∈ U
i
y
i
> 0
∆(z) =
1
y
i
z
n
=
y
n+1
y
i
y
n+1
=
z
n
∆(z)
(Dφ
+
i
)
y
(
X(y)) = y
m−1
n+1
(D(φ
+
i
◦ φ
+
))
x
X(x) =
z
m
n
(∆(z))
m−1
P
1
− z
1
P
i
, · · · , P
i+1
− z
i
P
i
, · · · , P
n
− z
n−1
P
i
, −z
n
P
i
=
1
(∆(z))
m−1
P
1
− z
1
P
i
, · · · ,
P
i+1
− z
i
P
i
, · · · ,
P
n
− z
n−1
P
i
, −z
n
P
i
,
P
j
=
P
j
(z
1
, ..., z
i−1
, 1, z
i
, ..., z
n
)
P
j
y ∈ U
i
∩ H
−
1 ≤ i ≤ n
φ
−
φ
+
(Dφ
+
i
)
y
(
X(y))
y ∈ U
n+1
= H
+
(Dφ
+
n+1
)
y
(
X(y)) = y
m−1
n+1
(D(φ
+
n+1
◦ φ
+
))
x
X(x)
y = φ
+
(x) φ
+
n+1
= (φ
+
)
−1
(D(φ
+
n+1
◦ φ
+
))
x
(z
1
, ..., z
n
) = φ
+
n+1
(y
1
, ..., y
n+1
) = φ
+
n+1
(φ
+
(x)) = (x
1
, ..., x
n
)
y
n+1
=
1
∆(x)
=
1
∆(z)
(Dφ
+
i
)
y
(
X(y)) = y
m−1
n+1
(P
1
, ..., P
n
) =
1
(∆(z))
m−1
(P
1
, ..., P
n
)
P
j
= P
j
(z
1
, ..., z
n
)
X
(∆(z))
m−1
> 0
X
U
1
P
2
− z
1
P
1
,
P
3
− z
2
P
1
, · · · ,
P
n
− z
n−1
P
1
, −z
n
P
1
P
j
=
P
j
(1, z
1
, ..., z
n
)
U
2
P
1
− z
1
P
2
,
P
3
− z
2
P
2
, · · · ,
P
n
− z
n−1
P
2
, −z
n
P
2
P
j
=
P
j
(z
1
, 1, z
2
, ..., z
n
)
U
n
P
1
− z
1
P
n
,
P
2
− z
2
P
n
, · · · ,
P
n−1
− z
n−1
P
n
, −z
n
P
n
P
j
=
P
j
(z
1
, ..., z
n
, 1)
U
n+1
P
1
, P
2
, ..., P
n
P
j
= P
j
(z
1
, z
2
, ..., z
n
)
X y ∈ V
i
φ
+
i
φ
−
i
U
i
(−1)
m−1
(z
1
, ..., z
n
) = φ
−
i
(y
1
, ..., y
n
) ∆(z) =
1
|y
i
|
= −
1
y
i
y
i
< 0 V
i
S
n−1
S
n
X ϕ
X
ϕ(t
0
) ∈ S
n−1
ϕ(t) ∈ S
n−1
t ϕ
ϕ = (ϕ
1
, ..., ϕ
n
)
X ϕ(t
0
) ∈ S
n−1
ϕ
X U
i
i = 1, ..., n
ϕ
′
n
(t) = ϕ
n
(t)
P
i
(ϕ
1
(t), ..., ϕ
n
(t)) ϕ(t
0
) = 0
ϕ
n
(t) = 0 ϕ(t) ∈ S
n−1
t
ϕ
V
i
i = 1, ..., n
S
n
n n S
n
X R
n
y ∈ S
n−1
X(y) = 0
X
X(y) = 0
y ∈ H
+
H
−
= S
n
\ S
n−1
n
X
S
n
X
S
n−1
n
n − 1
X|
S
n−1
X
S
n−1
R
n+1
S
n
R
n
Z = (S
1
, ..., S
n
, S
n+1
) R
n+1
Z|
S
n
S
n
m ∈ Z
+
∂
α
S
i
(0) = 0 |α| = m m + 2 i = 1, ..., n ∂
α
S
n+1
(0) = 0 |α| = m + 2
∂
α
S
i
(0) = 0 |α| = m + 2 α
i
= 0 α
i
= 1 α
n+1
= m + 1
i = 1, ..., n
∂
e
i
∂
α
S
i
(0)
(α
i
+ 1)
=
∂
e
j
∂
α
S
j
(0)
(α
j
+ 1)
,
|α| = m + 1 i, j = 1, ..., n
∂
e
i
∂
α
S
i
(0)
(α
i
+ 1)
= −
α
j
≥1
1≤j≤ n
α
j
∂
β
j
(α)
S
j
(0)
|α| = m + 1 i = 1, ..., n
β
j
(α) = α − e
j
|β
j
(α)| = m
Z
Z(y) =
1 − y
2
1
−y
1
y
2
... −y
1
y
n
−y
2
y
1
1 − y
2
2
... −y
2
y
n
−y
1
y
n
−y
2
y
n
... 1 − y
2
n
−y
1
y
n+1
−y
2
y
n+1
... −y
n
y
n+1
R
1
(y)
R
2
(y)
R
n
(y)
,
R
i
(y) =
|α|=m
∂
α
S
i
(0)
α!
y
α
, 1 ≤ i ≤ n
y = (y
1
, ..., y
n
, y
n+
1
) 0
R
n+1
∂
α
S
i
(0) = 0 |α| = m i 1 ≤ i ≤ n
Z|
S
n
S
n
S
i
≡ 0 i = 1, ...n, n + 1
Z ≡ 0 Z ≡ 0
α (n + 1) i 1 ≤ i ≤ n α
′
= α + e
i
α |α| = m + 1 α
i
≥ 1 i β
j
= β
j
(α)
∂
α
′
S
i
(0)
α
′
!
=
∂
e
i
∂
α
S
i
(0)
α
′
!
=
−(α
i
+ 1)
α
′
!
α
j
≥1
1≤j≤ n
α
j
∂
β
j
S
j
(0) = −
α
j
≥1
1≤j≤ n
∂
β
j
S
j
(0)
(β
j
)!
,
(α
i
+ 1)
α
′
!
=
1
α!
α
j
α!
= (β
j
)!
α
j
≥ 1
|α| = m + 1 α
i
= 0 α
n+1
= m + 1 α
′
! = α!
∂
α
′
S
i
(0)
α
′
=
∂
e
i
∂
α
S
i
(0)
α!
= −
1
α!
α
j
≥1
1≤j≤ n
j=i
α
j
∂
β
j
S
j
(0) = −
α
j
≥1
1≤j≤ n
j=i
∂
β
j
S
j
(0)
(β
j
)!
S
i
m m + 2 i 1 ≤ i ≤ n
S
i
(y) =
|α|=m
∂
α
S
i
(0)
α!
y
α
+
|α|=m+2
∂
α
S
i
(0)
α!
y
α
S
m+2
i
(y) m + 2 S
i
S
m+2
i
(y) =
|α|=m+2
α
i
≥1
∂
α
S
i
(0)
α!
y
α
=
|α|=m+2
α
i
≥2
∂
α
S
i
(0)
α!
y
α
+
|α|=m+2
α
i
=1
α
n+1
=m+1
∂
α
S
i
(0)
α!
y
α
S
m+2
i
(y) α
i
≥ 1
S
m+2
i
(y) = y
i
|α|=m+1
α
i
≥1
∂
α
′
S
i
(0)
α
′
!
y
α
+ y
i
|α|=m+1
α
i
=0
α
n+1
=m+1
∂
α
′
S
i
(0)
α
′
!
y
α
,
α
′
= α + e
i
|α|=m+1
α
i
≥1
α
j
≥1
1≤j≤ n
∂
β
j
S
j
(0)
(β
j
)!
y
α
=
|α|=m+1
α
i
≥1
∂
β
i
S
i
(0)
(β
i
)!
y
α
+
|α|=m+1
α
i
≥1
α
j
≥1
1≤j≤ n
j=i
∂
β
j
S
j
(0)
(β
j
)!
y
α
S
m+2
i
(y) = y
i
|α|=m+1
α
i
≥1
−
α
j
≥1
1≤j≤ n
∂
β
j
S
j
(0)
(β
j
)!
y
α
+ y
i
|α|=m+1
α
i
=0
α
n+1
=m+1
−
α
j
≥1
1≤j≤ n
j=i
∂
β
j
S
j
(0)
(β
j
)!
y
α
= −y
i
|α|=m+1
α
i
≥1
∂
β
i
S
i
(0)
(β
i
)!
y
α
− y
i
|α|=m+1
α
j
≥1
1≤j≤ n
j=i
∂
β
j
S
j
(0)
(β
j
)!
y
α
= −y
i
|α|=m+1
α
i
≥1
∂
β
i
S
i
(0)
(β
i
)!
y
α
− y
i
n
j=1
j=i
|α|=m+1
α
j
≥1
∂
β
j
S
j
(0)
(β
j
)!
y
α
y
i
≥ 1 y
j
≥ 1
β
k
(α) + e
k
= α k = 1, ..., n
S
m+2
i
(y) = −y
2
i
|α|=m
∂
α
S
i
(0)
α!
y
α
− y
i
y
j
n
j=1
j=i
|α|=m
∂
α
S
j
(0)
α!
y
α
R
k
(y) m S
k
(y) k = 1, ..., n
S
i
(y) = (1 − y
2
i
)
|α|=m
∂
α
S
i
(0)
α!
y
α
− y
i
y
j
n
j=1
j=i
|α|=m
∂
α
S
j
(0)
α!
y
α
= (1 − y
2
i
)R
i
(y) −
n
j=1
j=i
y
i
y
j
R
j
(y) ,
Z|
S
n
S
n
(y
1
, ..., y
n
, y
n+1
) ∈ S
n
n+1
i=1
y
i
S
i
(y) = 0
S
i
R
k
n+1
i=1
y
i
S
i
(y) =
n
i=1
y
i
(1 − y
2
i
)R
i
(y) −
n
j=1
j=i
y
i
y
j
R
j
(y)
+ y
n+1
S
n+1
(y) =
n
i=1
y
i
(1 − y
2
i
)R
i
(y) −
n
i=1
n
j=1
j=i
y
2
i
y
j
R
j
(y) + y
n+1
S
n+1
(y)
S
n
(1 − y
2
i
) =
n+1
j=1
j=i
y
2
j
n
i=1
y
i
(1 − y
2
i
)R
i
(y) =
n
i=1
n+1
j=1
j=i
y
i
y
2
j
R
i
(y) =
n+1
j=1
n
i=1
i=j
y
i
y
2
j
R
i
(y) =
n
j=1
n
i=1
i=j
y
i
y
2
j
R
i
(y) +
n
i=1
y
i
y
2
n+1
R
i
(y)
n+1
i=1
y
i
S
i
(y) =
n
i=1
y
i
y
2
n+1
R
i
(y) + y
n+1
S
n+1
(y)
(y
1
, ..., y
n
, y
n+1
) ∈ S
n
m + 3
n
i=1
y
i
y
2
n+1
R
i
(y) + y
n+1
S
n+1
(y) = 0
R
n+1
S
n+1
(y) = −
n
i=1
y
i
y
n+1
R
i
(y)
Z
Z(y) =
1 − y
2
1
−y
1
y
2
... −y
1
y
n
−y
2
y
1
1 − y
2
2
... −y
2
y
n
−y
1
y
n
−y
2
y
n
... 1 − y
2
n
−y
1
y
n+1
−y
2
y
n+1
... −y
n
y
n+1
R
1
(y)
R
2
(y)
R
n
(y)
,
R
n+1
R
n
Z = (S
1
, ..., S
n
, S
n+1
) R
n+1
∃ i, 1 ≤ i ≤ n, α, |α| = m, α
n+1
= 0 ∂
α
S
i
(0) = 0.
Z|
S
n
X = (P
1
, ..., P
n
) R
n
P
i
(x) =
|α|=m
m
α
n+1
=0
∂
α
S
i
(0)
α!
x
Π
n
(α)
x = (x
1
, ..., x
n
)
Π
n
(α) = (α
1
, ..., α
n
)
X = (P
1
, ..., P
n
) R
n
(X) = max{ (P
1
), ..., (P
n
)} = m
X(y) = y
m−1
n+1
X(y) =
1 − y
2
1
−y
1
y
2
... −y
1
y
n
−y
2
y
1
1 − y
2
2
... −y
2
y
n
−y
1
y
n
−y
2
y
n
... 1 − y
2
n
−y
1
y
n+1
−y
2
y
n+1
... −y
n
y
n+1
P
1
(y)
P
2
(y)
P
n
(y)
,
P
i
(y
1
, ..., y
n+1
) = y
m
n+1
P
i
(y
1
/y
n+1
, ..., y
n
/y
n+1
)
x
Π
n
(α)
= x
α
1
1
.....x
α
n
n
x = (x
1
, ..., x
n
) = (y
1
/y
n+1
, ..., y
n
/y
n+1
)
x
Π
n
(α)
= (
y
1
y
n+1
)
α
1
...(
y
n
y
n+1
)
α
n
= y
α
1
1
...y
α
n
n
y
(−
P
n
j=1
α
j
)
n+1
= y
α
1
1
...y
α
n
n
y
(−|α|+α
n+1
)
n+1
.
P
i
(y
1
, ..., y
n+1
) = y
m
n+1
P
i
(y
1
/y
n+1
, ..., y
n
/y
n+1
)
= y
m
n+1
|α|=m
m
α
n+1
=0
∂
α
S
i
(0)
α!
y
α
1
1
...y
α
n
n
y
(−m+α
n+1
)
n+1
=
|α|=m
∂
α
S
i
(0)
α!
y
α
= R
i
(y)
P
i
(y) = R
i
(y)
X = Z|
S
n
Z = (S
1
, S
2
, S
3
) R
3
S
1
(y) = y
1
y
3
+ y
2
y
3
− y
1
y
2
y
2
3
− y
2
1
y
2
y
3
− y
3
1
y
3
,
S
2
(y) = y
2
3
− y
2
1
y
2
y
3
− y
1
y
2
2
y
3
− y
2
2
y
2
3
S
3
(y) = −y
1
y
2
y
2
3
− y
2
1
y
2
3
− y
2
y
3
3
Z m = 2
n = 2
Z(y) =
1 − y
2
1
−y
1
y
2
−y
1
y
2
1 − y
2
2
−y
1
y
3
−y
2
y
3
R
1
(y)
R
2
(y)
,
R
1
(y) =
|α|=2
∂
α
S
1
(0)
α!
y
α
= y
1
y
3
+ y
2
y
3
R
2
(y) =
|α|=2
∂
α
S
2
(0)
α!
y
α
= y
2
3
|α| = 2 α
n+1
= 0 ∂
α
S
i
(0) = 0 i = 1, 2 Z
Z
X = (P
1
, P
2
) R
2
P
1
(x) =
|α|=2
2
α
n+1
=0
∂
α
S
1
(0)
α!
x
π
2
(α)
= x
1
+ x
2
P
2
(x) =
|α|=2
2
α
n+1
=0
∂
α
S
2
(0)
α!
x
π
2
(α)
= 1
Z = (S
1
, S
2
, S
3
) R
3
S
1
(y) = y
1
+ y
2
− y
3
1
− y
2
1
y
2
− y
1
y
2
y
3
,
S
2
(y) = y
3
− y
2
1
y
2
− y
1
y
2
2
− y
2
2
y
3
S
3
(y) = −y
1
y
2
y
3
− y
2
1
y
3
− y
2
y
2
3
Z m = 1
n = 2
Z(y) =
1 − y
2
1
−y
1
y
2
−y
1
y
2
1 − y
2
2
−y
1
y
3
−y
2
y
3
R
1
(y)
R
2
(y)
,
R
1
(y) =
|α|=1
∂
α
S
1
(0)
α!
y
α
= y
1
+ y
2
R
2
(y) =
|α|=1
∂
α
S
2
(0)
α!
y
α
= y
3
∂
α
S
1
(0) = 1 α = (1, 0, 0) (0, 1, 0)
Z
X = (P
1
, P
2
) R
2
P
1
(x) = x
1
+ x
2
P
2
(x) = 1
R
n+1
X R
n
X = (P
1
, ..., P
n
) R
n
(X) = m
X S
n
X = (S
1
, ..., S
n
, S
n+1
)
S
i
(y) = (1 − y
2
i
)
P
i
(y) −
n
j=1
j=i
y
i
y
j
P
j
(y),
S
n+1
(y) = −
n
i=1
y
i
y
n+1
P
i
(y)
P
k
X
y ∈ S
n
(1 − y
2
i
) =
n+1
j=1
j=i
y
2
j
X R
n
X
X(y) =
n
i=2
y
2
i
−y
1
y
2
... −y
1
y
n
−y
2
y
1
n
i=1
i=2
y
2
i
... −y
2
y
n
−y
1
y
n
−y
2
y
n
...
n−1
i=1
y
2
i
−y
1
y
n+1
−y
2
y
n+1
... −y
n
y
n+1
P
1
(y)
P
2
(y)
P
n
(y)
,
P
i
(y
1
, ..., y
n+1
) = y
m
n+1
P
i
(y
1
/y
n+1
, ..., y
n
/y
n+1
)
R
3
Z = (S
1
, S
2
, S
3
) R
3
Z|
S
2
S
2
m ∈ Z
∗
+
∂
α
S
i
(0) = 0 |α| = m + 2 i = 1, 2 , 3
∂
α
S
i
(0) = 0 |α| = m + 2 α
i
= m + 2 α
i
= m + 1 α
3
= 1 i = 1, 2
∂
e
2
∂
α
S
1
(0)
α
2
+ 1
=
−∂
e
1
∂
α
S
2
(0)
α
1
+ 1
,
|α| = m + 1 α
3
= 0 1
∂
α
S
1
(0)
α!
+
[
α
1
2
]
j=0
[
α
2
2
]
k=1
(−1)
j+k
f
j
(k − 1)
∂
β(j,k)
S
1
(0)
β(j, k)!
+
[
α
1
−1
2
]
j=0
[
α
2
−1
2
]
k=0
(−1)
j+k
f
j
(k)
∂
γ(j,k)
S
2
(0)
γ(j, k)!
= 0,
α α
3
= 0 1 1 ≤ α
1
≤ m α
2
≥ 2
f
j
(k) f
j
(0) = 1 j f
0
(k) = 1 k j, k ≥ 1
f
j
(k) = f
j−1
(k) + f
j
(k − 1)
β(j, k) = β(j, k, α) = α − 2j e
1
− 2k e
2
+ 2(j + k) e
3
γ(j, k) = γ(j, k, α) = α − (2 j + 1) e
1
− (2 k + 1) e
2
+ 2(j + k + 1) e
3
∂
α
S
i
(0)
α!
+
[
α
ℓ
2
]
k=1
(−1)
k
∂
β(0,k)
S
i
(0)
β(0, k)!
= 0,
i = 1, 2 ℓ ∈ {1, 2} ℓ = i α α
i
= 0 α
ℓ
≥ 2 α
3
= 0 1
β(0, k) = β(0, k, α) = α − 2k e
ℓ
+ 2k e
3
Z
Z(y) =
y
2
2
+ y
2
3
−y
1
y
2
−y
1
y
2
y
2
1
+ y
2
3
−y
1
y
3
−y
2
y
3
Q
1
(y)
Q
2
(y)
,
Q
i
(y) =
|α|=m
α
i
=0
[
α
′
ℓ
−1
2
]
k=0
(−1)
k
∂
γ(0,k)
S
i
(0)
γ(0, k) !
y
α
+
|α|=m
α
ℓ
=0
α
i
≥1
∂
γ(0,0)
S
i
(0)
γ(0, 0)!
y
α
+
|α|=m
α
ℓ
=1
α
i
≥1
∂
γ(0,0)
S
i
(0)
γ(0, 0)!
+
[
α
′
i
2
]
j=1
(−1)
j+1
∂
β(j,1)
S
ℓ
(0)
β(j, 1)!
y
α
+
|α|=m
α
i
≥1
α
ℓ
≥2
∂
γ(0,0)
S
i
(0)
γ(0, 0)!
+
[
α
′
i
−1
2
]
j=0
[
α
′
ℓ
−1
2
]
k=1
(−1)
j+k
f
j
(k − 1)
∂
γ(j,k)
S
i
(0)
γ(j, k)!
+
[
α
′
i
2
]
j=1
[
α
′
ℓ
2
]
k=1
(−1)
j+k
f
j−1
(k − 1)
∂
β(j,k)
S
ℓ
(0)
β(j, k)!
y
α
i, ℓ ∈ {1, 2} i = ℓ
β(j, k) = β(j, k, α
′
) = α
′
− 2j e
i
− 2k e
ℓ
+ 2(j + k) e
3
γ(j, k) = γ(j, k, α
′
) = α
′
− (2 j + 1) e
i
− (2 k + 1) e
ℓ
+ 2(j + k + 1) e
3
,
α
′
= α + e
i
+ e
ℓ
[ ]
[r] = r n ∈ Z
[r] + n = [r + n]
α m + 2 α
3
= 0 1 α
1
, α
2
≥ 1
∂
α
S
1
(0)
α!
=
∂
α
′
S
2
(0)
α
′
!
α
′
= α + e
1
− e
2
∂
α
S
2
(0)
α!
+
[
α
1
2
]
j=1
[
α
2
2
]
k=0
(−1)
j+k
f
j−1
(k)
∂
β(j,k)
S
2
(0)
β(j, k)!
+
[
α
1
−1
2
]
j=0
[
α
2
−1
2
]
k=0
(−1)
j+k
f
j
(k)
∂
γ(j,k)
S
1
(0)
γ(j, k)!
= 0,
α α
3
= 0 1 1 ≤ α
2
≤ m α
1
≥ 2
(4)
′
∂
α
S
i
(0)
α!
+
[
α
i
2
]
j=0
[
α
ℓ
2
]
k=1
(−1)
j+k
f
j
(k − 1)
∂
β(j,k)
S
i
(0)
β(j, k)!
+
[
α
i
−1
2
]
j=0
[
α
ℓ
−1
2
]
k=0
(−1)
j+k
f
j
(k)
∂
γ(j,k)
S
ℓ
(0)
γ(j, k)!
= 0,
i, ℓ ∈ {1, 2} i = ℓ α α
3
= 0 1 1 ≤ α
i
≤ m α
ℓ
≥ 2
β(j, k) = β(j, k, α) = α − 2j e
i
− 2k e
ℓ
+ 2(j + k) e
3
γ(j, k) = γ(j, k, α) = α − (2j + 1) e
i
− (2k + 1) e
ℓ
+ 2(j + k + 1) e
3
j, k ∈ Z
+
f
j
(k)
f
j
(0) = 1 j f
0
(k) = 1 k j, k ≥ 1 f
j
(k) = f
j−1
(k)+f
j
(k −1)
f
j
(k) = f
k
(j)
j k
j = 0 f
j
(k) f
0
(k) = f
k
(0) = 1 ∀ k ∈ Z
+
f
j
(k) = f
k
(j) ∀ k f
j+1
(k) = f
k
(j + 1) ∀ k
k = 0 f
j+1
(0) = f
0
(j + 1) = 1
f
j+1
(k) = f
k
(j + 1) f
j+1
(k + 1) = f
k+1
(j + 1)
f
j+1
(k+1) = f
j+1
(k)+f
j
(k+1) j k
f
j
(k+1) = f
k+1
(j) f
j+1
(k) = f
k
(j+1) f
j+1
(k+1) = f
k
(j+1)+f
k+1
(j) =
f
k+1
(j + 1)
f
j+1
(k) = f
k
(j + 1) ∀ k f
j
(k) = f
k
(j)
i ∈ {1, 2} ∂
α
S
i
(0) = 0
|α| = m + 2 α
i
= m + 2 α
i
= m + 1 α
3
= 1 S
i
S
i
(y) = S
(1,i)
(y) + S
(2,i)
(y) + S
(3,i)
(y),
S
(1,i)
(y) =
|α|=m+2
α
i
=0
α
ℓ
≥2
∂
α
S
i
(0)
α!
y
α
S
(2,i)
(y) =
|α|=m+2
α
i
=0
α
ℓ
≤1,α
3
≥2
∂
α
S
i
(0)
α!
y
α
+
|α|=m+2
α
i
≥1,α
3
≥2
∂
α
S
i
(0)
α!
y
α
S
(3,i)
(y) =
|α|=m+2
α
i
,α
ℓ
≥1
α
3
≤1
∂
α
S
i
(0)
α!
y
α
α 3 m + 2 ℓ ∈ {1, 2}
ℓ = i
β(j, k) = β(j, k, α) = α − 2j e
i
− 2k e
ℓ
+ 2(j + k) e
3
γ(j, k) = γ(j, k, α) = α − (2j + 1) e
i
− (2k + 1) e
ℓ
+ 2(j + k + 1) e
3
α
i
, α
ℓ
≥ 1
S
i
(I)
|α|=m+2
α
i
=0
α
ℓ
,α
3
≥2
[
α
ℓ
2
]
k=0
(−1)
k
∂
β(0,k)
S
i
(0)
β(0, k)!
y
α
(II)
|α|=m+2
α
ℓ
=0
α
i
≥1,α
3
≥2
∂
α
S
i
(0)
α!
y
α+2(e
ℓ
−e
3
)
(III)
|α|=m+2
α
ℓ
=1
α
i
≥1,α
3
≥2
∂
α
S
i
(0)
α!
+
[
α
i
−1
2
]
j=0
(−1)
j
∂
γ(j,0)
S
ℓ
(0)
γ(j, 0)!
y
α+2(e
ℓ
−e
3
)
(IV )
|α|=m+2
α
i
≥1
α
ℓ
,α
3
≥2
∂
α
S
i
(0)
α!
+
[
α
i
2
]
j=0
[
α
ℓ
2
]
k=1
(−1)
j+k
f
j
(k − 1)
∂
β(j,k)
S
i
(0)
β(j, k)!
+
[
α
i
−1
2
]
j=0
[
α
ℓ
−1
2
]
k=0
(−1)
j+k
f
j
(k)
∂
γ(j,k)
S
ℓ
(0)
γ(j, k)!
y
α+2(e
ℓ
−e
3
)
(V )
|α|=m+2
α
i
=0
α
ℓ
≥1,α
3
≥4
∂
α
S
ℓ
(0)
α!
y
α+(e
i
+e
ℓ
−2e
3
)
(V I)
|α|=m+2
α
ℓ
=0
α
3
≥4
[
α
i
2
]
j=0
(−1)
j
∂
β(j,0)
S
ℓ
(0)
β(j, 0)!
y
α+(e
i
+e
ℓ
−2e
3
)
(V II)
|α|=m+2
α
i
=1
α
ℓ
≥1,α
3
≥4
∂
α
S
ℓ
(0)
α!
+
[
α
ℓ
−1
2
]
k=0
(−1)
k
∂
γ(0,k)
S
i
(0)
γ(0, k) !
y
α+(e
i
+e
ℓ
−2e
3
)
(V III)
|α|=m+2
α
ℓ
≥1
α
i
≥2,α
3
≥4
∂
α
S
ℓ
(0)
α!
+
[
α
i
2
]
j=1
[
α
ℓ
2
]
k=0
(−1)
j+k
f
j−1
(k)
∂
β(j,k)
S
ℓ
(0)
β(j, k)!
+
[
α
i
−1
2
]
j=0
[
α
ℓ
−1
2
]
k=0
(−1)
j+k
f
j
(k)
∂
γ(j,k)
S
i
(0)
γ(j, k)!
y
α+(e
i
+e
ℓ
−2e
3
)
m m ≥ 1
m ≥ 2 m ≥ 3
m ≥ 4 m ≥ 5
β(j, k) + 2(e
3
− e
ℓ
) = β(j, k + 1) β(j, k) + (2e
3
− e
i
− e
ℓ
) = γ(j, k)
γ(j, k) + 2(e
3
− e
ℓ
) = γ(j, k + 1) γ(j, k) + (2e
3
− e
i
− e
ℓ
) = β(j + 1, k + 1)
α + 2(e
3
− e
ℓ
) = β(0, 1) α + (2e
3
− e
i
− e
ℓ
) = γ(0, 0)
(II)
′
|α|=m+2
α
ℓ
=2
α
i
≥1
∂
β(0,1)
S
i
(0)
β(0, 1)!
y
α
(III)
′
|α|=m+2
α
ℓ
=3
α
i
≥1
∂
β(0,1)
S
i
(0)
β(0, 1)!
+
[
α
i
−1
2
]
j=0
(−1)
j
∂
γ(j,1)
S
ℓ
(0)
γ(j, 1)!
y
α
(IV )
′
|α|=m+2
α
ℓ
≥4
α
i
≥1
∂
β(0,1)
S
i
(0)
β(0, 1)!
+
[
α
i
2
]
j=0
[
α
ℓ
2
]
k=2
(−1)
j+k+1
f
j
(k − 2)
∂
β(j,k)
S
i
(0)
β(j, k)!
+
[
α
i
−1
2
]
j=0
[
α
ℓ
−1
2
]
k=1
(−1)
j+k+1
f
j
(k − 1)
∂
γ(j,k)
S
ℓ
(0)
γ(j, k)!
y
α
(V )
′
|α|=m+2
α
i
=1
α
ℓ
,α
3
≥2
∂
γ(0,0)
S
ℓ
(0)
γ(0, 0)!
y
α
(V I)
′
|α|=m+2
α
ℓ
=1
α
i
≥1,α
3
≥2
[
α
i
−1
2
]
j=0
(−1)
j
∂
γ(j,0)
S
ℓ
(0)
γ(j, 0)!
y
α
(V II)
′
|α|=m+2
α
i
=2
α
ℓ
,α
3
≥2
∂
γ(0,0)
S
ℓ
(0)
γ(0, 0)!
+
[
α
ℓ
2
]
k=1
(−1)
k+1
∂
β(1,k)
S
i
(0)
β(1, k)!
y
α
(V III)
′
|α|=m+2
α
i
≥3
α
ℓ
,α
3
≥2
∂
γ(0,0)
S
ℓ
(0)
γ(0, 0)!
+
[
α
i
−1
2
]
j=1
[
α
ℓ
−1
2
]
k=0
(−1)
j+k
f
j−1
(k)
∂
γ(j,k)
S
ℓ
(0)
γ(j, k)!
+
[
α
i
2
]
j=1
[
α
ℓ
2
]
k=1
(−1)
j+k+2
f
j−1
(k − 1)
∂
β(j,k)
S
i
(0)
β(j, k)!
y
α
(II)
′′
, (III)
′′
(IV )
′′
(II)
′
, (III)
′
(IV )
′
α
3
≥ 2 (II) , (III) (IV ) α
3
≥ 4
(II)
′′′
, (III)
′′′
(IV )
′′′
(II)
′
, (III)
′
(IV )
′
α
3
≤ 1
(II), (III) (IV ) α
3
≤ 3 (II)
′
= (II)
′′
+ (II)
′′′
(III)
′
= (III)
′′
+ (III)
′′′
(IV )
′
= (IV )
′′
+ (IV )
′′′
S
i
(y) = R
(1,i)
(y) + R
(2,i)
(y) + R
(3,i)
(y),
R
(1,i)
(y) = S
(1,i)
(y) − (I) + (II) + (III) + (IV ),
R
(2,i)
(y) = S
(2,i)
(y) + (I) − (II)
′′
− (III)
′′
− (IV )
′′
+ (V )
′
+ (V I)
′
+ (V II)
′
+ (V III)
′
R
(3,i)
(y) = S
(3,i)
(y) − (II)
′′′
− (III)
′′′
− (IV )
′′′
− (V ) − (V I) − (V II) − (V III)
R
(1,i)
(y) =
|α|=m+2
α
i
=0
α
3
≤1
∂
α
S
i
(0)
α!
y
α
+
|α|=m+2
α
i
=0
α
ℓ
,α
3
≥2
[
α
ℓ
2
]
k=1
(−1)
k+1
∂
β(0,k)
S
i
(0)
β(0, k)!
y
α
+
|α|=m+2
α
ℓ
=0
α
i
≥1,α
3
≥2
∂
α
S
i
(0)
α!
y
α+2(e
ℓ
−e
3
)
+
|α|=m+2
α
ℓ
=1
α
i
≥1,α
3
≥2
∂
α
S
i
(0)
α!
+
[
α
i
−1
2
]
j=0
(−1)
j
∂
γ(j,0)
S
ℓ
(0)
γ(j, 0)!
y
α+2(e
ℓ
−e
3
)
+
+
|α|=m+2
α
i
≥1
α
ℓ
,α
3
≥2
∂
α
S
i
(0)
α!
+
[
α
i
2
]
j=0
[
α
ℓ
2
]
k=1
(−1)
j+k
f
j
(k − 1)
∂
β(j,k)
S
i
(0)
β(j, k)!
+
[
α
i
−1
2
]
j=0
[
α
ℓ
−1
2
]
k=0
(−1)
j+k
f
j
(k)
∂
γ(j,k)
S
ℓ
(0)
γ(j, k)!
y
α+2(e
ℓ
−e
3
)
R
(2,i)
(y) =
|α|=m+2
α
i
=0
α
3
≥2
[
α
ℓ
2
]
k=0
(−1)
k
∂
β(0,k)
S
i
(0)
β(0, k)!
y
α
+
|α|=m+2
α
ℓ
=0
α
i
≥1,α
3
≥2
∂
α
S
i
(0)
α!
y
α
+
|α|=m+2
α
ℓ
=1
α
i
≥1,α
3
≥2
∂
α
S
i
(0)
α!
+
[
α
i
−1
2
]
j=0
(−1)
j
∂
γ(j,0)
S
ℓ
(0)
γ(j, 0)!
y
α
+
|α|=m+2
α
i
≥1
α
ℓ
,α
3
≥2
∂
α
S
i
(0)
α!
+
[
α
i
2
]
j=0
[
α
ℓ
2
]
k=1
(−1)
j+k
f
j
(k − 1)
∂
β(j,k)
S
i
(0)
β(j, k)!
+
[
α
i
−1
2
]
j=0
[
α
ℓ
−1
2
]
k=0
(−1)
j+k
f
j
(k)
∂
γ(j,k)
S
ℓ
(0)
γ(j, k)!
y
α
R
(3,i)
(y) =
|α|=m+2
α
ℓ
=1
α
i
≥1,α
3
≤1
∂
α
S
i
(0)
α!
y
α
+
|α|=m+2
α
ℓ
=2
α
i
≥1,α
3
≤1
∂
α
S
i
(0)
α!
−
∂
β(0,1)
S
i
(0)
β(0, 1)!
y
α
+
|α|=m+2
α
ℓ
=3
α
i
≥1,α
3
≤1
∂
α
S
i
(0)
α!
−
∂
β(0,1)
S
i
(0)
β(0, 1)!
−
[
α
i
−1
2
]
j=0
(−1)
j
∂
γ(j,1)
S
ℓ
(0)
γ(j, 1)!
y
α
+
|α|=m+2
α
ℓ
≥4
α
i
≥1,α
3
≤1
∂
α
S
i
(0)
α!
−
∂
β(0,1)
S
i
(0)
β(0, 1)!
−
[
α
i
2
]
j=0
[
α
ℓ
2
]
k=2
(−1)
j+k+1
f
j
(k − 2)
∂
β(j,k)
S
i
(0)
β(j, k)!
−
[
α
i
−1
2
]
j=0
[
α
ℓ
−1
2
]
k=1
(−1)
j+k+1
f
j
(k − 1)
∂
γ(j,k)
S
ℓ
(0)
γ(j, k)!
y
α
−
|α|=m+2
α
i
=0
α
ℓ
≥1,α
3
≥4
∂
α
S
ℓ
(0)
α!
y
α+(e
i
+e
ℓ
−2e
3
)
−
|α|=m+2
α
ℓ
=0
α
3
≥4
[
α
i
2
]
j=0
(−1)
j
∂
β(j,0)
S
ℓ
(0)
β(j, 0)!
y
α+(e
i
+e
ℓ
−2e
3
)
−
|α|=m+2
α
i
=1
α
ℓ
≥1,α
3
≥4
∂
α
S
ℓ
(0)
α!
+
[
α
ℓ
−1
2
]
k=0
(−1)
k
∂
γ(0,k)
S
i
(0)
γ(0, k) !
y
α+(e
i
+e
ℓ
−2e
3
)
−
|α|=m+2
α
i
≥2
α
ℓ
≥1,α
3
≥4
∂
α
S
ℓ
(0)
α!
+
[
α
i
2
]
j=1
[
α
ℓ
2
]
k=0
(−1)
j+k
f
j−1
(k)
∂
β(j,k)
S
ℓ
(0)
β(j, k)!
+
[
α
i
−1
2
]
j=0
[
α
ℓ
−1
2
]
k=0
(−1)
j+k
f
j
(k)
∂
γ(j,k)
S
i
(0)
γ(j, k)!
y
α+(e
i
+e
ℓ
−2e
3
)
R
(1,i)
R
(2,i)
y
ℓ
y
3
2 R
(3,i)
y
1
y
2
1
α |α| = m α
′
= α+e
i
+e
ℓ
α+2e
ℓ
= α−e
i
+e
ℓ
β(0, k, α)+2e
ℓ
= γ(0, k −1, α
′
) α+2e
3
= γ(0, 0, α
′
) γ(j, k, α)+2 e
3
= β(j+1, k+1, α
′
)
β(j, k, α) + 2e
3
= γ(j, k, α
′
)
R
(1,i)
(y) = y
2
ℓ
Q
(1,i)
(y),
R
(2,i)
(y) = y
2
3
Q
(2,i)
(y)
R
(3,i)
(y) = −y
1
y
2
Q
(3,i)
(y),
Q
(1,i)
(y) =
|α|=m
α
i
=0
α
3
≤1
∂
α
′
−e
i
+e
ℓ
S
i
(0)
(α
′
− e
i
+ e
ℓ
)!
y
α
+
|α|=m
α
i
=0
α
3
≥2
[
α
′
ℓ
−1
2
]
k=0
(−1)
k
∂
γ(0,k)
S
i
(0)
γ(0, k) !
y
α
+
|α|=m
α
ℓ
=0
α
i
≥1
∂
γ(0,0)
S
i
(0)
γ(0, 0)!
y
α
+
|α|=m
α
ℓ
=1
α
i
≥1
∂
γ(0,0)
S
i
(0)
γ(0, 0)!
+
[
α
′
i
2
]
j=1
(−1)
j+1
∂
β(j,1)
S
ℓ
(0)
β(j, 1)!
y
α
+
|α|=m
α
ℓ
≥2
α
i
≥1
∂
γ(0,0)
S
i
(0)
γ(0, 0)!
+
[
α
′
i
−1
2
]
j=0
[
α
′
ℓ
−1
2
]
k=1
(−1)
j+k
f
j
(k − 1)
∂
γ(j,k)
S
i
(0)
γ(j, k)!
+
[
α
′
i
2
]
j=1
[
α
′
ℓ
2
]
k=1
(−1)
j+k
f
j−1
(k − 1)
∂
β(j,k)
S
ℓ
(0)
β(j, k)!
y
α
Q
(2,i)
(y) =
|α|=m
α
i
=0
[
α
′
ℓ
−1
2
]
k=0
(−1)
k
∂
γ(0,k)
S
i
(0)
γ(0, k) !
y
α
+
|α|=m
α
ℓ
=0
α
i
≥1
∂
γ(0,0)
S
i
(0)
γ(0, 0)!
y
α
+
|α|=m
α
ℓ
=1
α
i
≥1
∂
γ(0,0)
S
i
(0)
γ(0, 0)!
+
[
α
′
i
2
]
j=1
(−1)
j+1
∂
β(j,1)
S
ℓ
(0)
β(j, 1)!
y
α
+
+
|α|=m
α
ℓ
≥2
α
i
≥1
∂
γ(0,0)
S
i
(0)
γ(0, 0)!
+
[
α
′
i
−1
2
]
j=0
[
α
′
ℓ
−1
2
]
k=1
(−1)
j+k
f
j
(k − 1)
∂
γ(j,k)
S
i
(0)
γ(j, k)!
+
[
α
′
i
2
]
j=1
[
α
′
ℓ
2
]
k=1
(−1)
j+k
f
j−1
(k − 1)
∂
β(j,k)
S
ℓ
(0)
β(j, k)!
y
α
Q
(3,i)
(y) = −
|α|=m
α
ℓ
=0
α
3
≤1
∂
α
′
S
i
(0)
α
′
!
y
α
+
|α|=m
α
ℓ
=1
α
3
≤1
−
∂
α
′
S
i
(0)
α
′
!
+
∂
β(0,1)
S
i
(0)
β(0, 1)!
y
α
+
|α|=m
α
ℓ
=2
α
3
≤1
−
∂
α
′
S
i
(0)
α
′
!
+
∂
β(0,1)
S
i
(0)
β(0, 1)!
+
[
α
′
i
−1
2
]
j=0
(−1)
j
∂
γ(j,1)
S
ℓ
(0)
γ(j, 1)!
y
α
+
|α|=m
α
ℓ
≥3
α
3
≤1
−
∂
α
′
S
i
(0)
α
′
!
+
∂
β(0,1)
S
i
(0)
β(0, 1)!
+
[
α
′
i
2
]
j=0
[
α
′
ℓ
2
]
k=2
(−1)
j+k+1
f
j
(k − 2)
∂
β(j,k)
S
i
(0)
β(j, k)!
+
[
α
′
i
−1
2
]
j=0
[
α
′
ℓ
−1
2
]
k=1
(−1)
j+k+1
f
j
(k − 1)
∂
γ(j,k)
S
ℓ
(0)
γ(j, k)!
y
α
+
|α|=m
α
i
=0
α
ℓ
≥1,α
3
≥2
∂
γ(0,0)
S
ℓ
(0)
γ(0, 0)!
y
α
+
|α|=m
α
ℓ
=0
α
3
≥2
[
α
′
i
−1
2
]
j=0
(−1)
j
∂
γ(j,0)
S
ℓ
(0)
γ(j, 0)!
y
α
+
|α|=m
α
i
=1
α
ℓ
≥1,α
3
≥2
∂
γ(0,0)
S
ℓ
(0)
γ(0, 0)!
+
[
α
′
ℓ
2
]
k=1
(−1)
k+1
∂
β(1,k)
S
i
(0)
β(1, k)!
y
α
+
+
|α|=m
α
ℓ
≥1
α
i
,α
3
≥2
∂
γ(0,0)
S
ℓ
(0)
γ(0, 0)!
+
[
α
′
i
−1
2
]
j=1
[
α
′
ℓ
−1
2
]
k=0
(−1)
j+k
f
j−1
(k)
∂
γ(j,k)
S
ℓ
(0)
γ(j, k)!
+
[
α
′
i
2
]
j=1
[
α
′
ℓ
2
]
k=1
(−1)
j+k
f
j−1
(k − 1)
∂
β(j,k)
S
i
(0)
β(j, k)!
y
α
β(j, k) = β(j, k, α
′
) γ(j, k) = γ(j, k, α
′
) Q
(1,i)
Q
(2,i)
Q
(3,i)
γ(0, k, α
′
) = β(0, k + 1, α
′
− e
i
+ e
ℓ
)
Q
(1,i)
= Q
(2,i)
Q
(2,ℓ)
(y) =
|α|=m
α
ℓ
=0
[
α
′
i
−1
2
]
j=0
(−1)
j
∂
γ(j,0)
S
ℓ
(0)
γ(j, 0)!
y
α
+
|α|=m
α
i
=0
α
ℓ
≥1
∂
γ(0,0)
S
ℓ
(0)
γ(0, 0)!
y
α
+
|α|=m
α
i
=1
α
ℓ
≥1
∂
γ(0,0)
S
ℓ
(0)
γ(0, 0)!
+
[
α
′
ℓ
2
]
k=1
(−1)
k+1
∂
β(1,k)
S
i
(0)
β(1, k)!
y
α
+
|α|=m
α
i
≥2
α
ℓ
≥1
∂
γ(0,0)
S
ℓ
(0)
γ(0, 0)!
+
[
α
′
ℓ
−1
2
]
k=0
[
α
′
i
−1
2
]
j=1
(−1)
j+k
f
k
(j − 1)
∂
γ(j,k)
S
ℓ
(0)
γ(j, k)!
+
[
α
′
ℓ
2
]
k=1
[
α
′
i
2
]
j=1
(−1)
j+k
f
k−1
(j − 1)
∂
β(j,k)
S
i
(0)
β(j, k)!
y
α
Q
(3,i)
= Q
(2,ℓ)
= Q
(1,ℓ)
Q
(1,i)
= Q
(2,i)
=
Q
(3,ℓ)
Q
(1,i)
= Q
(2,i)
= Q
(3,ℓ)
= Q
i
Q
(1,ℓ)
= Q
(2,ℓ)
= Q
(3,i)
= Q
ℓ
i = 1, 2
S
i
(y) = R
(1,i)
(y) + R
(2,i)
(y) + R
(3,i)
(y) = (y
2
ℓ
+ y
2
3
) Q
i
(y) − y
1
y
2
Q
ℓ
(y)
Z|
S
2
S
2
y ∈ S
2
y
1
S
1
+ y
2
S
2
+ y
3
S
3
= 0
S
1
S
2
y
1
S
1
+ y
2
S
2
+ y
3
S
3
= y
1
y
2
3
Q
1
+ y
2
y
2
3
Q
2
+ y
3
S
3
(y
1
, y
2
, y
3
) ∈ S
2
m + 3
S
3
(y) = −y
1
y
3
Q
1
− y
2
y
3
Q
2
R
3
Z
Z(y) =
y
2
2
+ y
2
3
−y
1
y
2
−y
1
y
2
y
2
1
+ y
2
3
−y
1
y
3
−y
2
y
3
Q
1
(y)
Q
2
(y)
,
Q
i
Q
i
i = 1, 2
Q
i
(y) =
|α|=m
∂
α
Q
i
(0)
α!
y
α
Q
1
Q
2
y
3
0 Z X = (P
1
, P
2
)
P
i
(x) =
|α|=m
m
α
3
=0
∂
α
Q
i
(0)
α!
x
α
1
1
x
α
2
2
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